As just reported in The Age (and The SMH), the Grattan Institute has a new report out, apparently concluding that a lot of primary mathematics teaching sucks. Well duh, and more power to Grattan. Taking Grattan’s lead, the Age report also gives Clarendon College a strong plug, which is predictable and maybe fine. (There is a disconcerting ring to “lesson plans that have been honed over several years”.) Similarly, the Age report gives Bentleigh West a strong plug, which is predictable and which we presume is fine.
These are the good guys, or at minimum the significantly better guys. We hope to give the Grattan report a careful read in the next day or so, and we’re keen to learn more about both CC and BW. But, if there is a WitCH to be done then there is a WitCH to be done. And so here we are.
The Age article indicates some reasons for the good guys’ success (to the extent that NAPLAN results are a measure of success): explicit teaching, of course; weekly reviews; “a sequential way of teaching maths” (raising the disturbing question of what a non-sequential way might look like); the incorporation of specialist mathematics teachers. None of that, however, provides any proper clue of the mathematics in these classrooms, the facts and techniques that the students would be expected to learn, and the type of exercises and problems that the students will tackle. To be fair, it is not easy to provide much of a picture in a newspaper article. Still, without some such detail an article can only be sketchy, and just one sort-of-a-clue is provided in the Age article.
Below is a problem that the The Age included, from a year 2 Bentleigh West “worksheet”. It includes “answers”, as shown (which were presumably not included on the actual worksheet).
We have no idea how this problem was chosen for the article, and such illustration material tends to be chosen with less care than that given to the body of the article. Still, it is presumably considered to be a good problem. Perhaps it is a good problem. That is far from obvious to us.
My feelings about problems like the one illustrated is that it is not suitable for regular classroom use, but if you want to put it on a bulletin board and kids can come along and fill in their suggestions intermittently, then fine. Something might appear that would be worth following up in class.
Let’s face it — classes cannot be exciting all the time and a lot of straightforward important but routine stuff needs to be done. But it is worth teaching children in classes because of the human interractions and the things that can change the dynamic and lead to something memorable, especially if the teacher is opportunistic in seeing the opportunities.
It is like families eating together. Most meals are pretty much the same day by day, but it provides the opportunity for something important to happen.
Thanks, Ed. Can you indicate why you regard problems “like” this one as not suitable for regular classroom use?
(Note, we don’t know how BW used the problem, except that it was supposedly on a Y2 “worksheet”.)
How many ways/times can you make 23…
It depends.
Am I only allowed to use the numbers given in the grid?
Can I use a number more than once?
Do
, 
and 
all count as one “time” or are they different?
There is no mention of whether or not I can use brackets, but the answer uses them (even though they are not required in these answers, but that is another matter entirely).
Also, the “answers” have missed
and possibly other “times” as well.
Thanks, RF. You’re subsequent questions are a little (but not a lot) nitpicky, but your first question seems on point. I don’t think it’s great when the answer to a Year 2 question is “infinity”.
Hm. What if it said ‘Give four different ways of making 23’? Then the student would know when they can move on.
Thanks, aps. That would intrinsically be a better problem although not remotely good enough, and also seemingly against the intention of the activity.
If you look at the Grattan report, p 53, it includes the exact same type (and a preferable version) of the problem as an “early finisher task” for Year 4. Presumably the problem above is used by BW in the exact same way in Year 2. So, having lots of answers to find is intrinsic to it use, and restricting to four answers or whatever is defeating the purpose.
It is understandable to have an early finisher task that is not so black or white as solved or unsolved. You don’t (necessarily) want the kid to spend ten minutes on the thing and then have it unsolved. (But they could always take the thing away …). This is where you have to look closely at what is set (and why I distrust AI to create such sheets, or teachers to check them for aptness).
Look at the actual target and the actual supplied numbers. There are a ton of ways to get to 23 just using addition, even without re-using numbers as The Age did. What is the point of having that many solutions? What is the point of asking for solutions involving multiplication if the kid can easily avoid it? How many kids are going to use division? Why 23 as a target?
Now, the activity may work out better than that in practice, because it’s the smarter and/or motivated kids who finish early, and they may challenge themselves, or be nudged by the teacher. But why leave it to the chance sense and wonder of the kid and the teacher? Why not simply have such a task but significantly more directed, to have the kid engage better with the ideas and calculations appropriate for the kid and the specific level and topic at hand?
For example, it would seem to me that it would be much much much better to provide a target, say 24, and then provide the exact numbers to be used, three or four of them, each of them to be used exactly once. You could keep the same target and then the exercises could be various choices of numbers, with the puzzles getting increasingly harder: the early ones could require just addition and have multiple possibilities, working up to later ones that require division or are tricky in subtle ways.
How would that not be way, way better than the BW sheet?
I can see this could possibly be a nice class exercise, with students volunteering to go to the board and write one solution. At first I thought the question was bizarre, since this kind of question is usually posed in a way that there are only finitely many solutions. But on reflection, maybe this is for young kids and part of the idea is that the teacher can celebrate the genius of the first student who realizes there are infinitely many solutions?
Thanks, Grant. Maybe you’re agreeing with Ed, then, even if that’s a bit beside the point. Note that the question was on a “worksheet” rather than a group exercise or the like.
As it happens, I doubt that “infinity” was the genius answer aimed for. The Grattan report has an essentially identical but differently worded question, and with the target 87, as a Year 4 BW “fast finisher activity”.
In any case, and even given the aspect that you and Ed point out, I really don’t like the problem.
Yes, I agree it’s not a good worksheet problem. But maybe ok for a class problem, provided the teacher is excited by such things. And for older students, the Frobenius coin problem is a wonderful related problem, and important too.
Even as a class problem, I think it is poor.
I guess it could be nice for the classroom from point of view that students can approach it in a simple or sophisticated way. But you are leaving it up to chance for students to learn something meaningful from it. For a year 2 student, any insights they make would be from a hunch or guess. I’m not sure how much that would be worth to their learning.
Thanks, Potii. I agree. This is one amongst a number of reasons why I dislike the exercise.
This reminds me of Tony Gardiner’s example of the “24 game” : make 24 using 3, 3, 4, 4 (or 3, 3, 5, 5 etc.) (Gardiner, 2016, p. 13). He says that it is an example of an ‘inverse’ challenge and contrasts it with normal ‘direct’ calculations that are often the only maths that students are exposed to. He emphasises that “in the long run, it is the inverse operation which is generally more important.” (Gardiner, 2016, p. 14).
With your particular example, I’m glad that there are addition and subtraction solutions as year 2 students are not yet learning their times tables.
Gardiner, T. (2016). Teaching Mathematics at Secondary Level (1st ed., Vol. 2). Open Book Publishers. https://doi.org/10.11647/OBP.0071
Thanks, MW. Yes, The game of 24 naturally comes to mind. I’ve written about 24 and Gardiner’s really excellent puzzle, here. An update at the end of the post is a long reflection on the game by Tony Gardiner. And, yes, the importance of inverse problems is difficult to overestimate. But.
It is not properly comparing like for like, but the game of 24 and Tony’s puzzle are far, far superior to BW’s problem.
I read the report from Grattan Institute. I wish that commentators would use the word “mathematics” instead of “maths” in public documents.
As for other recent public comments, I do not like commentators quoting one very low ATAR entry point to judge a course in teaching. I like to say that Australia is a very poor country because I know a bloke who has almost no money.
Also, using ATAR scores to judge education courses seems to ignore the fact that prospective secondary teachers are usually accepted for MTeach courses on the basis of having a university degree rather than an ATAR score.
Thanks, Terry. On “maths”, I agree: this vernacularisation is really gauche and really annoying, and it is becoming very common.
On the ATAR thing, was that in the Grattan report, or are you just tacking it on? Of course the nitpicking on the one poor bloke is nasty and evidence of little or nothing. But is it true that most teachers go through MTeach? Whether or not that is the case, are you really suggesting there is no issue with the strength and subject grounding of primary teachers and mathematics teachers?
I was just tacking on the ATAR thing.
I’d say that all new aspiring teachers are expected to complete the MTeach.
My point about prospective primary teachers and prospective secondary teachers was that they come into teaching from different pathways. This is often forgotten in ATAR debates.
But I don’t want to distract readers from your focus on the Grattan report.
If you ask a seven year old “How fast can you run to that tree and back?”, you are not asking for their best estimate, given the biomechanics involved. You are telling them to get sprinting.
This question is not asking how many solutions there are. It’s asking kids to have a crack at generating some. The “how many?” framing is so they don’t stop once they’ve got a few.
The point is for them to practice their mental number bonds (“what numbers multiply to 23? Oh, none. Ok. What numbers add to 23? Any of these? Nope. Alright can I generate any of those somehow?” Etc).
This is useful practice, adds variety from forward-direction calculation drills, and creates context transfer.
I don’t think this is at all ambiguous to anyone in grade two, teacher or student.
Thanks, Alex. I have this nitpicky thing, where I believe that a mathematics question should ask what it intends to ask, not something merely in the ballpark.
If nothing else, the wording of this exercise is crappy. Sure, one will aim for gentle loose wording for a year 2 question, and will permit some looseness, but BW can do much better than they did.
Pretty much every commenter here has pointed out the value of posing inverse problems, and thus the value of questions *like* the one here. That does not make the posted question a good example of the type. It is not, or even close.
When I saw the flooding coverage on my LinkedIn feed, the first thing that came to mind was to visit this blog and read Marty’s thoughts. Admittedly, it seems to me that many scholars today have a kind of explicit-teaching-phobia…
Jiquing Sun, The Victorian Department of Education places a strong emphasis on explicit teaching, as does my school which is a government school. Best wishes, Terry
Jiqing was talking about “scholars”.